4. Let 2x2 + y2 – 3xy = 0 be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

7. Let b ¹ 0 and for j = 0, 1, 2, …….., n, let Sj be the area of the region bounded

by the y-axis and the curve xeay = sin by, jπ/b ≤ y ≤ ((j+1)π)/b.

Show that S0, S1, S2, ………., Sn are in geometric progression. Also, find their sum for a = – 1 and b = π.

8. Let αÎ R. Prove that a function f : R --> R is differentiable at α if and only if there is a function g : R --> R which is continuous at α and satisfies f(x) – f(α) = f(x) (x – α) for all x Î R.

9. Let C1 and C2 be two circles with C2 lying inside C1. A circle C lying inside C1touches C1 internally and C2 externally. Identify the locus of the centre of C.

10. Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

12. (a) Let P be a point on the ellipse x2/a2 + y2/b2 =1, 0 < b < a. Let the line parallel to y-axis passing through P meet the circle x2 + y2 = a2 at the point Q such that P and Q are on the same side of x-axis. For two positive real numbers r and s, find the locus of the point R on PQ such that PR : RQ = r : s as P varies over the ellipse.

(b) If D is the area of a triangle with side lengths a, b, c then

show that D < 1/4 √((a+b+c)abc).

Also show that the equality occurs in the above inequality if and only if a = b = c.

13. A hemispherical tank of radius 2 metres is initially full of water and has an outlet of 12 cm2 cross-sectional are at the bottom. The outlet is opened at some instant.

The flow through the outlet is according to the law v(t) = 0.6 √(2gh(t)), where v(t) and h(t) are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank.
(Hint : Form a differential equation by relating the decrease of water level to the outflow).

14. (a) An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white?
(b) An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6 is thrown n times and the list of n numbers showing up is noted. What is the probability that, among the numbers 1, 2, 3, 4, 5, 6 only three numbers appear in this list?